Congressional leaders opened their doors Wednesday to former White House Chief of Staff Erskine Bowles and former Sen. Alan K. Simpson, R-Wyo., who were touting the debt reduction plan that bears their names as Capitol Hill gets closer and closer to the fiscal cliff.

But their efforts will be immediately overshadowed Thursday as high-level Obama administration officials meet with the top four congressional leaders on Capitol Hill.

The Bowles-Simpson plan, which calls for lowering tax rates while ending many tax deductions and overhauling entitlements, has been roundly panned by leaders on both sides since it was shot down nearly two years ago by the very fiscal commission that produced it.

But leaders hoping to be perceived as more open to a bipartisan deal — and, in effect, hoping to gain an eventual upper hand in negotiations — were eager to be seen meeting with one or both of the men.

“The plan symbolizes something greater than the specifics of the plan; the plan symbolizes compromise,” a GOP leadership aide said. “At some point I do think the speaker and president are going to get in a room, but a significant part of leverage on both sides comes from the public perception of the debate.”

The day when Speaker John A. Boehner, R-Ohio, and President Barack Obama meet about the fiscal cliff might be nearing.

Congressional leaders have expressed pessimism in recent days about the progress of the fiscal cliff talks, with both McConnell and Reid saying negotiators remain at loggerheads.

The visits from the top White House figures could help smooth over recent bumps, including GOP frustration that the president is hitting the road Friday to barnstorm for a deal instead of coming to the table.

“We accepted this meeting with the expectation that the White House team will bring a specific plan for real spending cuts — because spending cuts that Washington Democrats will accept is what is missing from the ‘balanced approach’ that the president says he wants,” a Boehner spokesman said.

Reid said he was not aware whether Geithner would be bringing a debt plan on Thursday. “I don’t know what he’s going to do,” Reid said of Geithner.

Majority Whip Richard J. Durbin of Illinois, meantime, said he hoped Geithner would have a plan. “He’s been thinking about it a long time,” Durbin said. “I hope he brings something that will bring us together.”

McConnell’s office said it was eager to see something. “We are eager to hear the administration’s specific plans for protecting jobs and growth while reducing the national debt through strengthening the entitlement system, reducing Washington spending and preventing a tax hike on every American taxpayer,” McConnell spokesman John Ashbrook said in a statement. “As Sen. McConnell made clear: ‘We can do this. But the president has to lead. That’s the issue here. It’s that simple.’”

That would seem to leave Simpson and Bowles, who have been making the rounds in Washington hard of late, out in the cold despite the fact that they’re being held up as standard-bearers of bipartisanship.

“They’re red-hot right now. Members want their pictures taken with them,” said Rep. Jim Cooper, D-Tenn., an early and ardent evangelist for the Bowles-Simpson plan.

Cooper met with the duo Wednesday afternoon along with a bipartisan group of legislators, many of whom were among the 38 members who voted for a budget modeled after Bowles-Simpson. Those members call themselves the “Brave 38.”

On Wednesday, Bowles and Simpson also held a press availability with House Democratic leaders, and Bowles met behind closed doors with Republican House leaders and a group of CEOs dedicated to debt reduction.

Bowles acknowledged the plan he and Simpson labored over in 2010 and are still touting is a tough sell on Capitol Hill. Coming out of his meeting with GOP leaders, he said the plan’s ideals are more important than the specifics.

“I think a balanced approach is a good starting point, and that’s what we’ve got to have. What we put out was a balanced approach, but it’s only one plan. There are lots of ways to get there,” he said. “I think if you ask anybody, from the minority leader to the speaker, if there are more people that favor the kind of balanced approach that we presented, they say, ‘Yes.’”

The problem remains, however, that while most lawmakers are calling for a balanced deal, they back away at the mention of specifics, said Rep. Peter Welch, D-Vt.

“Their plan has lots to criticize, but it has the one structural approach that everyone knows ultimately will be necessary,” Welch said. “The challenge now is to get specific.”

Bowles said that he sees no give from Democrats in the area of entitlement reform and similarly none from Republicans in raising tax rates.

Still, Cooper said the chances that a Bowles-Simpson type of plan can pass have “gone up dramatically,” saying “the whole world has changed,” since he and Rep. Steven C. LaTourette, R-Ohio, brought the budget to the floor.

“Simpson and Bowles were never wedded to every detail of their plan,” he said. “For the rest of us, it’s one of the few bipartisan ways to solve the problem.”

The four closed-door meetings on Thursday take place one day after Obama delivered remarks at the White House about the negotiations and a day before he hits the road to hold a campaign-style event in Pennsylvania to tout middle-class tax cuts.

“If both parties agree we should not raise taxes on middle-class families, let’s begin our work with where we agree,” Obama said Wednesday.

Geithner and Nabors also will meet with House Majority Leader Eric Cantor, R-Va., House Ways and Means Chairman Dave Camp, R-Mich., and House Budget Chairman Paul D. Ryan, R-Wis.